Question

A homogeneous rod $XY$ of length $L$ and mass $M$ is pivoted at the centre $C$ such that it can rotate freely in the vertical plane. Initially, the rod is in the horizontal position. A blob of wax of same mass $M$ as that of the rod falls vertically with the speed $V$ and sticks to the rod midway between points $C$ and $Y$. If the rod rotates with angular speed $\omega$ what will be angular speed in terms of $V$ and $L$?

Answer 1

By law of conservation of angular momentum,

$L_{wax}$ about C$=I_{\omega }$ system about C

$\Rightarrow MV\frac{L}{4}=[\frac{M{L}^2}{12}+M(\frac{L}{4}{)}^2]\omega \Rightarrow \frac{VL}{4}=[\frac{L^2}{12}+\frac{L^2}{16}]\omega \Rightarrow \omega =\frac{12v}{7L}$